In [1]}, [2]}, the first author proved that
\( (\mathcal {E}, \mathcal {D}_{\circ }^{\mu })\)  is closable on \( L^2(\mathfrak {S},\mu )\) ,
and that its closure \( (\mathcal {E}, \mathcal {D})\)  is a quasi-regular Dirichlet form.
Here, \( \mathcal {D}_{\circ }\)  is the set consisting of local and smooth functions on \( \mathfrak {S}\) .
We take \( \mathcal {D}_{\circ }^{\mu }\)  such that
\( & \mathcal {D}_{\circ }^{\mu } = \lbrace  f \in \mathcal {D}_{\circ }\, ;\, \mathcal {E} ( f , f ) < \infty ,\, f \in L^2(\mathfrak {S},\mu )\rbrace .\) 
